To convert the given expression into its correct radical form, we need to understand how to handle the fractional exponent. The given expression is:
[tex]\[
\left(\frac{p^{12} q^{\frac{3}{2}}}{64}\right)^{\frac{5}{6}}
\][/tex]
Step-by-step process:
1. Understand the Fractional Exponent:
A fractional exponent [tex]\( \frac{m}{n} \)[/tex] implies taking the [tex]\( n \)[/tex]-th root of the base and then raising the result to the [tex]\( m \)[/tex]-th power. The fractional exponent here is [tex]\( \frac{5}{6} \)[/tex].
2. Rewrite the Expression:
We can rewrite the expression by first interpreting the fractional exponent:
[tex]\[
\left(\frac{p^{12} q^{\frac{3}{2}}}{64}\right)^{\frac{5}{6}} = \left(\sqrt[6]{\frac{p^{12} q^{\frac{3}{2}}}{64}}\right)^5
\][/tex]
3. Check the Options Provided:
We need to match our derived expression with one of the given options:
- A: [tex]\(\left(\sqrt[6]{\frac{p^{12} q^{\frac{3}{2}}}{64}}\right)^5\)[/tex]
- B: [tex]\(\left(\sqrt[5]{\frac{p^{12} q^{\frac{3}{2}}}{64}}\right)^6\)[/tex]
- C: [tex]\(\sqrt{\left(\frac{p^{12} q^{\frac{3}{2}}}{64}\right)^{\frac{5}{6}}}\)[/tex]
- D: [tex]\(\sqrt{\left(\frac{p^{12} q^{\frac{3}{2}}}{64}\right)^{\frac{1}{6}}}\)[/tex]
4. Compare and Identify the Correct Form:
Only option A matches the form that we derived.
[tex]\[
\left(\sqrt[6]{\frac{p^{12} q^{\frac{3}{2}}}{64}}\right)^5
\][/tex]
Therefore, the correct radical form of the expression is given in option A.
